# Full text of "Treatise On Analysis Vol-Ii"

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9   THE PLANCHEREL-GODEMENT THEOREM      377

To prove this, observe that for each G e 3fc(Sg) we have mg(GF) = mGF(l)
and, by virtue of (15.9.2.10), WGF = G  WF; hence TWGF(!) = JG(%) dmF(x);

but since mGF(l) = JG(x)F(%) dmg(x) by definition, this implies the relation
(15.9.2.14). In particular, bearing in mind (15.9.2.12), we have

(15.9.2.15)                               mx ,,=*]> w,

for all x, y in A. Since mx>y is a bounded measure, this relation implies that
jcj) is mg'integrable (13.14.4). The functions x therefore all belong to
, mg) ; also, it follows from (15.9.2.15) and (15.9.2.5) that, for all z e A,

= f

(15.9.2.16)        g(zx9 y) =    ffo) dmx,y(X) =

In other words, we have proved (15.9.2.1) in the particular case where x is
replaced by & product zx. It remains to show that (15.9.2.1) is true in general.
The remark at the beginning of (15.9.2.3) shows that, for all operators
KG.*;,

(15.9.2.17)        (V - ng(x) 1 7i,GO) = f

JS'g

in particular, taking V = lHg, we have

(15.9.2.18)                          9(x,y)
But, from (15.9.2.15), we have

f

jsg

Since the measure mXty is by definition induced by fj,Xty on Sg , it follows that
what we have to prove is that, when 0 e SŁ (and therefore Sg = S^  {0})

(15.9.2.19)                                ^({0}) = 0.

Now it follows immediately from the definition of ^x>y that the mapping
(x, y) i > {j,Xt y is sesquilinear, and hence (13.16.1) the complex-valued func-
tion (x, jO^/^./W) is a sesquilinear form on A x A. Moreover, it follows
directly from (15.9.2.17) and the Gelfand-Neumark theorem (15.4.14) that,
for each continuous function F on the compact space S^ = S^ u {0},

IK'

Uor each function F e <J>, we have